Old Problems and New Results in Coding Theory

Alexander Vardy
University of California, San Diego

Coding theory was born in 1948 with the work of Claude Shannon, who proved
that for every information rate R up to channel capacity, there exists a code
of rate R that guarantees a vanishing probability of decoding error. Shannon,
however, did not tell us how to find such codes nor how to decode them.

It was recognized early on that codes with good Hamming distance can correct
many errors, while codes endowed with algebraic structure admit efficient
algebraic decoding algorithms. This has led to over 50 years of research in
algebraic and combinatorial coding theory. We will survey several key problems
and new results in this area. In particular, we'll elaborate upon the recent
methods for decoding Reed-Solomon codes using bivariate polynomial interpolation.

About 10 years ago, the field of coding theory was transformed by the discovery
of codes defined on certain graphs, with no algebraic structure, that perform
extremely close to the Shannon capacity under probabilistic message-passing
decoding. We will briefly review this exciting development, and point out
the challenges that lie ahead in the area of "probabilistic" coding
theory.